Identifying the <em>x</em>- and <em>y</em>- Intercepts On a Graph

Identifying the x- and y- Intercepts on a Graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x- axis and the y- axis. These points are called the intercepts of the line.

Intercepts of a Line

The points where a line crosses the x- axis and the y- axis are called the intercepts of a line.

Let’s look at the graphs of the lines in the figure below.

Four figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line crossing the x- axis at the point (3, 0) and crossing the y- axis at the point (0, 6). The graph is labeled with the equation 2x plus y equals 6. Figure b shows a straight line crossing the x- axis at the point (4, 0) and crossing the y- axis at the point (0, negative 3). The graph is labeled with the equation 3x minus 4y equals 12. Figure c shows a straight line crossing the x- axis at the point (5, 0) and crossing the y- axis at the point (0, negative 5). The graph is labeled with the equation x minus y equals 5. Figure d shows a straight line crossing the x- axis and y- axis at the point (0, 0). The graph is labeled with the equation y equals negative 2x.

Examples of graphs crossing the x-negative axis.

First, notice where each of these lines crosses the \(x\) negative axis. See the figure above.

Figure	The line crosses the x- axis at:	Ordered pair of this point
Figure (a)	3	\(\left(3,0\right)\)
Figure (b)	4	\(\left(4,0\right)\)
Figure (c)	5	\(\left(5,0\right)\)
Figure (d)	0	\(\left(0,0\right)\)

Do you see a pattern?

For each row, the y- coordinate of the point where the line crosses the x- axis is zero. The point where the line crosses the x- axis has the form \(\left(a,0\right)\) and is called the x- intercept of a line. The x- intercept occurs when \(y\) is zero.

Now, let’s look at the points where these lines cross the y- axis. See the table below.

Figure	The line crosses the y-axis at:	Ordered pair for this point
Figure (a)	6	\(\left(0,6\right)\)
Figure (b)	\(-3\)	\(\left(0,-3\right)\)
Figure (c)	\(-5\)	\(\left(0,5\right)\)
Figure (d)	0	\(\left(0,0\right)\)

What is the pattern here?

In each row, the x- coordinate of the point where the line crosses the y- axis is zero. The point where the line crosses the y- axis has the form \(\left(0,b\right)\) and is called the y- intercept of the line. The y- intercept occurs when \(x\) is zero.

x- intercept and y- intercept of a line

The x- intercept is the point \(\left(a,0\right)\) where the line crosses the x- axis.

The y- intercept is the point \(\left(0,b\right)\) where the line crosses the y- axis.

Example

Find the x- and y- intercepts on each graph.

Three figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line going through the points (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), and (6, negative 1). Figure b shows a straight line going through the points (0, negative 6), (1, negative 3), (2, 0), (3, 3), and (4, 6). Figure c shows a straight line going through the points (negative 6, 1), (negative 5, 0), (negative 4, negative 1), (negative 3, negative 2), (negative 2, negative 3), (negative 1, negative 4), (0, negative 5), and (1, negative 6).

Solution

The graph crosses the x- axis at the point \(\left(4,0\right)\). The x- intercept is \(\left(4,0\right)\).
The graph crosses the y- axis at the point \(\left(0,2\right)\). The y- intercept is \(\left(0,2\right)\).
The graph crosses the x- axis at the point \(\left(2,0\right)\). The x- intercept is \(\left(2,0\right)\)
The graph crosses the y- axis at the point \(\left(0,-6\right)\). The y- intercept is \(\left(0,-6\right)\).
The graph crosses the x- axis at the point \(\left(-5,0\right)\). The x- intercept is \(\left(-5,0\right)\).
The graph crosses the y- axis at the point \(\left(0,-5\right)\). The y- intercept is \(\left(0,-5\right)\).